=================================================================================

You need to have more samples to make the bounds meaningful because the relationship between of the sample size (the number of data points) and the reliability or meaningfulness of statistical bounds, especially in the context of machine learning and statistical analysis. Here's why having more samples is often associated with more meaningful bounds:

1. Improved Generalization: In machine learning, one of the primary goals is to build models that can generalize well to unseen data. To estimate the generalization performance accurately, you need to have a sufficiently large and representative sample of data. More data points provide a better approximation of the true underlying data distribution, allowing you to make more reliable statements about how well your model will perform on new, unseen data.

2. Reduced Variance: As the sample size increases, the variance of your estimates tends to decrease. In other words, with more data, your statistical estimates become more stable and less sensitive to random fluctuations or noise in the data. This reduced variance leads to more robust and meaningful statistical bounds.

3. Confidence Intervals: Many statistical bounds, such as confidence intervals, rely on the sample size to determine their width. A larger sample size often results in narrower confidence intervals, indicating higher confidence in the accuracy of your estimates. Narrower intervals provide more precise information about the parameter being estimated.

4. Statistical Significance: In hypothesis testing and statistical analysis, having a larger sample size can lead to more statistically significant results. When the sample size is small, it's easier to encounter situations where observed differences or effects are due to random chance (sampling variability). With a larger sample, you are more likely to detect true effects or differences while minimizing the impact of random fluctuations.

5. Reduced Risk of Overfitting: In the context of model evaluation and selection, having more data reduces the risk of overfitting. Overfitting occurs when a model captures noise in the data rather than the true underlying patterns. With more data, the model is less likely to overfit, and the estimated bounds on its performance are more meaningful for unseen data.

However, note that the relationship between sample size and meaningful bounds can vary depending on the specific statistical analysis or machine learning task. In some cases, having a very large sample may not provide significant additional benefits in terms of bounds, especially if the data collection process is subject to systematic biases or if the data is inherently noisy.

In practice, determining an appropriate sample size is often a critical aspect of experimental design and statistical analysis. Researchers and data scientists aim to strike a balance between data availability, computational resources, and the level of precision required for their analysis to ensure that the bounds derived from their data are both meaningful and reliable.

============================================

The script below can be used to demonstrate the idea that having more samples can lead to more meaningful statistical bounds. We can visualize how the confidence intervals become narrower with larger sample sizes, indicating more meaningful bounds. We specify different sample sizes to test. On the other hadn, this script also shows the visualization of the relationship between sample size and the width of confidence intervals (bounds), which shows how the bounds change as a function of sample size. That is, the top plot shows how the mean estimate changes with varying sample sizes, while the bottom plot demonstrates how the width of the confidence intervals (bounds) changes as a function of sample size. Code:
         
       Output:    
         
         

The plot above shows the larger the sample size, the better the bounds.

============================================